7.3 Ballistic pendulum
A ballistic pendulum is a device that used to be used to determine the velocity of a bullet; it was invented in 1742 by English mathematician Benjamin Robins for this purpose. A simple version consists of a block of wood that hangs on a rod (the mass of the rod is small enough compared to the block to be ignored). You fire a bullet into the block, into which the bullet embeds itself. The block of wood, with the bullet lodged in it, then swings to some height that you can measure.
This device makes an excellent case study in conservation laws: there are two stages to this situation, each using different physical principles. The first stage is an inelastic collision between the bullet and block, in which momentum is conserved. After that, the bullet/block system has some velocity—so it has kinetic energy—that is converted into gravitational potential energy as the pendulum swings up.
Note: in the previous example, the y in the third line of the energy analysis should be an h. In the fourth line of the momentum analysis, the vw+b should not be there any more: it is entirely replaced by the expression √(2gh). The final calculation is correct.
Practice with ballistic pendulum-type problems
The ballistic pendulum is the prototypical example of a particular type of problem. These are problems involving a collision and also some process that comes after the collision. The collision involves conservation of energy, and the process that comes afterwards involves conservation of energy.
“Ballistic pendulum-type” problems are those where you need to consider both conservation of energy and momentum in different phases of a multi-step problem.